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Diffstat (limited to 'gcc-4.4.3/libstdc++-v3/include/tr1/modified_bessel_func.tcc')
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diff --git a/gcc-4.4.3/libstdc++-v3/include/tr1/modified_bessel_func.tcc b/gcc-4.4.3/libstdc++-v3/include/tr1/modified_bessel_func.tcc new file mode 100644 index 000000000..7b8aebc3b --- /dev/null +++ b/gcc-4.4.3/libstdc++-v3/include/tr1/modified_bessel_func.tcc @@ -0,0 +1,436 @@ +// Special functions -*- C++ -*- + +// Copyright (C) 2006, 2007, 2008, 2009 +// Free Software Foundation, Inc. +// +// This file is part of the GNU ISO C++ Library. This library is free +// software; you can redistribute it and/or modify it under the +// terms of the GNU General Public License as published by the +// Free Software Foundation; either version 3, or (at your option) +// any later version. +// +// This library is distributed in the hope that it will be useful, +// but WITHOUT ANY WARRANTY; without even the implied warranty of +// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +// GNU General Public License for more details. +// +// Under Section 7 of GPL version 3, you are granted additional +// permissions described in the GCC Runtime Library Exception, version +// 3.1, as published by the Free Software Foundation. + +// You should have received a copy of the GNU General Public License and +// a copy of the GCC Runtime Library Exception along with this program; +// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see +// <http://www.gnu.org/licenses/>. + +/** @file tr1/modified_bessel_func.tcc + * This is an internal header file, included by other library headers. + * You should not attempt to use it directly. + */ + +// +// ISO C++ 14882 TR1: 5.2 Special functions +// + +// Written by Edward Smith-Rowland. +// +// References: +// (1) Handbook of Mathematical Functions, +// Ed. Milton Abramowitz and Irene A. Stegun, +// Dover Publications, +// Section 9, pp. 355-434, Section 10 pp. 435-478 +// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl +// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky, +// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992), +// 2nd ed, pp. 246-249. + +#ifndef _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC +#define _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC 1 + +#include "special_function_util.h" + +namespace std +{ +namespace tr1 +{ + + // [5.2] Special functions + + // Implementation-space details. + namespace __detail + { + + /** + * @brief Compute the modified Bessel functions @f$ I_\nu(x) @f$ and + * @f$ K_\nu(x) @f$ and their first derivatives + * @f$ I'_\nu(x) @f$ and @f$ K'_\nu(x) @f$ respectively. + * These four functions are computed together for numerical + * stability. + * + * @param __nu The order of the Bessel functions. + * @param __x The argument of the Bessel functions. + * @param __Inu The output regular modified Bessel function. + * @param __Knu The output irregular modified Bessel function. + * @param __Ipnu The output derivative of the regular + * modified Bessel function. + * @param __Kpnu The output derivative of the irregular + * modified Bessel function. + */ + template <typename _Tp> + void + __bessel_ik(const _Tp __nu, const _Tp __x, + _Tp & __Inu, _Tp & __Knu, _Tp & __Ipnu, _Tp & __Kpnu) + { + if (__x == _Tp(0)) + { + if (__nu == _Tp(0)) + { + __Inu = _Tp(1); + __Ipnu = _Tp(0); + } + else if (__nu == _Tp(1)) + { + __Inu = _Tp(0); + __Ipnu = _Tp(0.5L); + } + else + { + __Inu = _Tp(0); + __Ipnu = _Tp(0); + } + __Knu = std::numeric_limits<_Tp>::infinity(); + __Kpnu = -std::numeric_limits<_Tp>::infinity(); + return; + } + + const _Tp __eps = std::numeric_limits<_Tp>::epsilon(); + const _Tp __fp_min = _Tp(10) * std::numeric_limits<_Tp>::epsilon(); + const int __max_iter = 15000; + const _Tp __x_min = _Tp(2); + + const int __nl = static_cast<int>(__nu + _Tp(0.5L)); + + const _Tp __mu = __nu - __nl; + const _Tp __mu2 = __mu * __mu; + const _Tp __xi = _Tp(1) / __x; + const _Tp __xi2 = _Tp(2) * __xi; + _Tp __h = __nu * __xi; + if ( __h < __fp_min ) + __h = __fp_min; + _Tp __b = __xi2 * __nu; + _Tp __d = _Tp(0); + _Tp __c = __h; + int __i; + for ( __i = 1; __i <= __max_iter; ++__i ) + { + __b += __xi2; + __d = _Tp(1) / (__b + __d); + __c = __b + _Tp(1) / __c; + const _Tp __del = __c * __d; + __h *= __del; + if (std::abs(__del - _Tp(1)) < __eps) + break; + } + if (__i > __max_iter) + std::__throw_runtime_error(__N("Argument x too large " + "in __bessel_jn; " + "try asymptotic expansion.")); + _Tp __Inul = __fp_min; + _Tp __Ipnul = __h * __Inul; + _Tp __Inul1 = __Inul; + _Tp __Ipnu1 = __Ipnul; + _Tp __fact = __nu * __xi; + for (int __l = __nl; __l >= 1; --__l) + { + const _Tp __Inutemp = __fact * __Inul + __Ipnul; + __fact -= __xi; + __Ipnul = __fact * __Inutemp + __Inul; + __Inul = __Inutemp; + } + _Tp __f = __Ipnul / __Inul; + _Tp __Kmu, __Knu1; + if (__x < __x_min) + { + const _Tp __x2 = __x / _Tp(2); + const _Tp __pimu = __numeric_constants<_Tp>::__pi() * __mu; + const _Tp __fact = (std::abs(__pimu) < __eps + ? _Tp(1) : __pimu / std::sin(__pimu)); + _Tp __d = -std::log(__x2); + _Tp __e = __mu * __d; + const _Tp __fact2 = (std::abs(__e) < __eps + ? _Tp(1) : std::sinh(__e) / __e); + _Tp __gam1, __gam2, __gampl, __gammi; + __gamma_temme(__mu, __gam1, __gam2, __gampl, __gammi); + _Tp __ff = __fact + * (__gam1 * std::cosh(__e) + __gam2 * __fact2 * __d); + _Tp __sum = __ff; + __e = std::exp(__e); + _Tp __p = __e / (_Tp(2) * __gampl); + _Tp __q = _Tp(1) / (_Tp(2) * __e * __gammi); + _Tp __c = _Tp(1); + __d = __x2 * __x2; + _Tp __sum1 = __p; + int __i; + for (__i = 1; __i <= __max_iter; ++__i) + { + __ff = (__i * __ff + __p + __q) / (__i * __i - __mu2); + __c *= __d / __i; + __p /= __i - __mu; + __q /= __i + __mu; + const _Tp __del = __c * __ff; + __sum += __del; + const _Tp __del1 = __c * (__p - __i * __ff); + __sum1 += __del1; + if (std::abs(__del) < __eps * std::abs(__sum)) + break; + } + if (__i > __max_iter) + std::__throw_runtime_error(__N("Bessel k series failed to converge " + "in __bessel_jn.")); + __Kmu = __sum; + __Knu1 = __sum1 * __xi2; + } + else + { + _Tp __b = _Tp(2) * (_Tp(1) + __x); + _Tp __d = _Tp(1) / __b; + _Tp __delh = __d; + _Tp __h = __delh; + _Tp __q1 = _Tp(0); + _Tp __q2 = _Tp(1); + _Tp __a1 = _Tp(0.25L) - __mu2; + _Tp __q = __c = __a1; + _Tp __a = -__a1; + _Tp __s = _Tp(1) + __q * __delh; + int __i; + for (__i = 2; __i <= __max_iter; ++__i) + { + __a -= 2 * (__i - 1); + __c = -__a * __c / __i; + const _Tp __qnew = (__q1 - __b * __q2) / __a; + __q1 = __q2; + __q2 = __qnew; + __q += __c * __qnew; + __b += _Tp(2); + __d = _Tp(1) / (__b + __a * __d); + __delh = (__b * __d - _Tp(1)) * __delh; + __h += __delh; + const _Tp __dels = __q * __delh; + __s += __dels; + if ( std::abs(__dels / __s) < __eps ) + break; + } + if (__i > __max_iter) + std::__throw_runtime_error(__N("Steed's method failed " + "in __bessel_jn.")); + __h = __a1 * __h; + __Kmu = std::sqrt(__numeric_constants<_Tp>::__pi() / (_Tp(2) * __x)) + * std::exp(-__x) / __s; + __Knu1 = __Kmu * (__mu + __x + _Tp(0.5L) - __h) * __xi; + } + + _Tp __Kpmu = __mu * __xi * __Kmu - __Knu1; + _Tp __Inumu = __xi / (__f * __Kmu - __Kpmu); + __Inu = __Inumu * __Inul1 / __Inul; + __Ipnu = __Inumu * __Ipnu1 / __Inul; + for ( __i = 1; __i <= __nl; ++__i ) + { + const _Tp __Knutemp = (__mu + __i) * __xi2 * __Knu1 + __Kmu; + __Kmu = __Knu1; + __Knu1 = __Knutemp; + } + __Knu = __Kmu; + __Kpnu = __nu * __xi * __Kmu - __Knu1; + + return; + } + + + /** + * @brief Return the regular modified Bessel function of order + * \f$ \nu \f$: \f$ I_{\nu}(x) \f$. + * + * The regular modified cylindrical Bessel function is: + * @f[ + * I_{\nu}(x) = \sum_{k=0}^{\infty} + * \frac{(x/2)^{\nu + 2k}}{k!\Gamma(\nu+k+1)} + * @f] + * + * @param __nu The order of the regular modified Bessel function. + * @param __x The argument of the regular modified Bessel function. + * @return The output regular modified Bessel function. + */ + template<typename _Tp> + _Tp + __cyl_bessel_i(const _Tp __nu, const _Tp __x) + { + if (__nu < _Tp(0) || __x < _Tp(0)) + std::__throw_domain_error(__N("Bad argument " + "in __cyl_bessel_i.")); + else if (__isnan(__nu) || __isnan(__x)) + return std::numeric_limits<_Tp>::quiet_NaN(); + else if (__x * __x < _Tp(10) * (__nu + _Tp(1))) + return __cyl_bessel_ij_series(__nu, __x, +_Tp(1), 200); + else + { + _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; + __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); + return __I_nu; + } + } + + + /** + * @brief Return the irregular modified Bessel function + * \f$ K_{\nu}(x) \f$ of order \f$ \nu \f$. + * + * The irregular modified Bessel function is defined by: + * @f[ + * K_{\nu}(x) = \frac{\pi}{2} + * \frac{I_{-\nu}(x) - I_{\nu}(x)}{\sin \nu\pi} + * @f] + * where for integral \f$ \nu = n \f$ a limit is taken: + * \f$ lim_{\nu \to n} \f$. + * + * @param __nu The order of the irregular modified Bessel function. + * @param __x The argument of the irregular modified Bessel function. + * @return The output irregular modified Bessel function. + */ + template<typename _Tp> + _Tp + __cyl_bessel_k(const _Tp __nu, const _Tp __x) + { + if (__nu < _Tp(0) || __x < _Tp(0)) + std::__throw_domain_error(__N("Bad argument " + "in __cyl_bessel_k.")); + else if (__isnan(__nu) || __isnan(__x)) + return std::numeric_limits<_Tp>::quiet_NaN(); + else + { + _Tp __I_nu, __K_nu, __Ip_nu, __Kp_nu; + __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); + return __K_nu; + } + } + + + /** + * @brief Compute the spherical modified Bessel functions + * @f$ i_n(x) @f$ and @f$ k_n(x) @f$ and their first + * derivatives @f$ i'_n(x) @f$ and @f$ k'_n(x) @f$ + * respectively. + * + * @param __n The order of the modified spherical Bessel function. + * @param __x The argument of the modified spherical Bessel function. + * @param __i_n The output regular modified spherical Bessel function. + * @param __k_n The output irregular modified spherical + * Bessel function. + * @param __ip_n The output derivative of the regular modified + * spherical Bessel function. + * @param __kp_n The output derivative of the irregular modified + * spherical Bessel function. + */ + template <typename _Tp> + void + __sph_bessel_ik(const unsigned int __n, const _Tp __x, + _Tp & __i_n, _Tp & __k_n, _Tp & __ip_n, _Tp & __kp_n) + { + const _Tp __nu = _Tp(__n) + _Tp(0.5L); + + _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; + __bessel_ik(__nu, __x, __I_nu, __K_nu, __Ip_nu, __Kp_nu); + + const _Tp __factor = __numeric_constants<_Tp>::__sqrtpio2() + / std::sqrt(__x); + + __i_n = __factor * __I_nu; + __k_n = __factor * __K_nu; + __ip_n = __factor * __Ip_nu - __i_n / (_Tp(2) * __x); + __kp_n = __factor * __Kp_nu - __k_n / (_Tp(2) * __x); + + return; + } + + + /** + * @brief Compute the Airy functions + * @f$ Ai(x) @f$ and @f$ Bi(x) @f$ and their first + * derivatives @f$ Ai'(x) @f$ and @f$ Bi(x) @f$ + * respectively. + * + * @param __n The order of the Airy functions. + * @param __x The argument of the Airy functions. + * @param __i_n The output Airy function. + * @param __k_n The output Airy function. + * @param __ip_n The output derivative of the Airy function. + * @param __kp_n The output derivative of the Airy function. + */ + template <typename _Tp> + void + __airy(const _Tp __x, + _Tp & __Ai, _Tp & __Bi, _Tp & __Aip, _Tp & __Bip) + { + const _Tp __absx = std::abs(__x); + const _Tp __rootx = std::sqrt(__absx); + const _Tp __z = _Tp(2) * __absx * __rootx / _Tp(3); + + if (__isnan(__x)) + return std::numeric_limits<_Tp>::quiet_NaN(); + else if (__x > _Tp(0)) + { + _Tp __I_nu, __Ip_nu, __K_nu, __Kp_nu; + + __bessel_ik(_Tp(1) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); + __Ai = __rootx * __K_nu + / (__numeric_constants<_Tp>::__sqrt3() + * __numeric_constants<_Tp>::__pi()); + __Bi = __rootx * (__K_nu / __numeric_constants<_Tp>::__pi() + + _Tp(2) * __I_nu / __numeric_constants<_Tp>::__sqrt3()); + + __bessel_ik(_Tp(2) / _Tp(3), __z, __I_nu, __K_nu, __Ip_nu, __Kp_nu); + __Aip = -__x * __K_nu + / (__numeric_constants<_Tp>::__sqrt3() + * __numeric_constants<_Tp>::__pi()); + __Bip = __x * (__K_nu / __numeric_constants<_Tp>::__pi() + + _Tp(2) * __I_nu + / __numeric_constants<_Tp>::__sqrt3()); + } + else if (__x < _Tp(0)) + { + _Tp __J_nu, __Jp_nu, __N_nu, __Np_nu; + + __bessel_jn(_Tp(1) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); + __Ai = __rootx * (__J_nu + - __N_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); + __Bi = -__rootx * (__N_nu + + __J_nu / __numeric_constants<_Tp>::__sqrt3()) / _Tp(2); + + __bessel_jn(_Tp(2) / _Tp(3), __z, __J_nu, __N_nu, __Jp_nu, __Np_nu); + __Aip = __absx * (__N_nu / __numeric_constants<_Tp>::__sqrt3() + + __J_nu) / _Tp(2); + __Bip = __absx * (__J_nu / __numeric_constants<_Tp>::__sqrt3() + - __N_nu) / _Tp(2); + } + else + { + // Reference: + // Abramowitz & Stegun, page 446 section 10.4.4 on Airy functions. + // The number is Ai(0) = 3^{-2/3}/\Gamma(2/3). + __Ai = _Tp(0.35502805388781723926L); + __Bi = __Ai * __numeric_constants<_Tp>::__sqrt3(); + + // Reference: + // Abramowitz & Stegun, page 446 section 10.4.5 on Airy functions. + // The number is Ai'(0) = -3^{-1/3}/\Gamma(1/3). + __Aip = -_Tp(0.25881940379280679840L); + __Bip = -__Aip * __numeric_constants<_Tp>::__sqrt3(); + } + + return; + } + + } // namespace std::tr1::__detail +} +} + +#endif // _GLIBCXX_TR1_MODIFIED_BESSEL_FUNC_TCC |